On the Complexity of Search Problems George Pierrakos Mostly based on: On the Complexity of the Parity Argument and Other Insufficient Proofs of Existence

On the Complexity of Search Problems

George Pierrakos

Mostly based on: On the Complexity of the Parity Argument and Other Insufficient Proofs of Existence [Pap94] On total functions, existence theorems and computational complexity [MP91] How easy is local search? [JPY88] Computational Complexity [Pap92] The complexity of computing a Nash equilibrium [DGP06] The complexity of pure Nash equilibria [FPT04] slides and scribe notes from many people…

1TFNP and LeafCovering

Outline1. Generally on Search Problems

2. The Class TFNP

3. Subclasses of TFNP part I: PPA, PPAD Problems in PPA, PPAD Completeness in PPAD

4. Subclasses of TFNP part II: PPP, PLS

5. PPAD-completeness of NASH & the complexity of computing equilibria in congestion games



2. The Class TFNP





Decision Problems vs Search (or “function”) Problems

SAT Input: boolean CNF-formula φ Output: “yes” or “no”

FSAT Input: boolean CNF-formula φ Output: satisfying assignment or “no” if none exist


Are search problems harder?They are definitely not easier:

a poly-time algorithm for FSAT can be easily tweaked to give a poly-time algorithm for SAT

…and vice versa, FSAT “reduces” to SAT:

we can figure out a satisfying assignment by running poly-time algorithm for SAT n-times


The Classes FP and FNP L € NP iff

there exists poly-time computable RL(x,y) s.t.

X € L y { |y| ≤ p(|x|) & RL(x,y) }

Note how RL defines the problem-language L

The corresponding search problem ΠR(L) € FNP is:given an x find any y s.t. RL(x,y) and reply “no” if none exist

FSAT € FNP… what about FTSP? Are all FNP problems self-reducible like FSAT? [open?]

FP is the subclass of FNP where we only consider problems for which a poly-time algorithm is known


Reductions and completeness A function problem ΠR reduces to a function

problem ΠS if there exist log-space computable string functions f and g, s.t.

R(x,g(y)) S(f(x),y)

intuitively f reduces problem ΠR to ΠS and g transforms a solution of ΠS to one of ΠR

Standard notion of completeness works fine…


FP <?> FNP A proof a-la-Cook shows that FSAT is FNP-complete

Hence, if FSAT € FP then FNP = FP

But we showed self-reducibility for SAT, so the theorem follows:

Theorem: FP = FNP iff P=NP

So, why care for function problems anyway??



2. The Class TFNP





On total “functions”: the class TFNP What happens if the relation R is total?

i.e., for each x there is at least one y s.t. R(x,y)

Define TFNP to be the subclass of FNP where the relation R is total TFNP contains problems that always have a solution, e.g.

factoring, fix-point theorems, graph-theoretic problems, … How do we know a solution exists?

By an “inefficient proof of existence”, i.e. non-(efficiently)-constructive proof

The idea is to categorize the problems in TFNP based on the type of inefficient argument that guarantees their solution


Basic stuff about TFNP1. FP TFNP FNP

2. TFNP = F(NP coNP) NP = problems with “yes” certificate y s.t. R1(x,y) coNP = problems with “no” certificate z s.t. R2(x,y) for TFNP F(NP coNP) take R = R1 U R2 for F(NP coNP) TFNP take R1 = R and R2 = ø

3. There is an FNP-complete problem in TFNP iff NP = coNP : If NP = coNP then trivially FNP = TFNP : If the FNP-complete problem ΠR is in TFNP then:

FSAT reduces to ΠR via f and g, hence any unsatisfiable formula φ has a “no” certificate y, s.t. R(f(φ),y) (y exists since ΠR is in TFNP) and g(y)=“no”

4. TFNP is a semantic complexity class no complete problems! note how telling whether a relation is total is undecidable (and not even RE!!)


Documents

On the Complexity of Search Problems George Pierrakos Mostly based on: On the Complexity of the Parity Argument and Other Insufficient Proofs of Existence